Default Risk and Aggregate Fluctuations in Emerging Economies

Transcription

1 Default Risk and Aggregate Fluctuations in Emerging Economies Cristina Arellano University of Minnesota Federal Reserve Bank of Minneapolis First Version: November 2003 This Version: February 2005 Abstract Sovereign default in emerging countries is accompanied by interest rates spikes, deep recessions, and sharp real exchange rate devaluations. This paper develops a two sector small open economy model to study default risk and its interaction with output, consumption and real exchange rates. Default probabilities and interest rates depend on incentives for repayment. Default occurs in equilibrium because asset markets are incomplete. The model predicts that default incentives and default premia are higher in recessions, as observed in the data. The reason is that in a recession, a risk averse borrower finds it more costly to repay non-contingent debt and is more likely to default. In a quantitative exercise, the model matches several features of the Argentinian economy. The model can account for the recent default episode and the dynamics observed prior to default: higher interest rate premia, capital outflows, real exchange rate depreciation, and collapses in consumption. An important implications of the model is that economies with relatively small tradable sectors have higher incentives to default on dollar denominated debt and thus have larger default probabilities. JEL Classification: E44, F32, F34 Keywords: Sovereign Debt, Default, Interest Rates, Real Exchange Rates I would like to thank my advisors Enrique Mendoza and Jonathan Heathcote for their guidance. I also thank Patrick Kehoe, John Coleman, Adam Szeidl, Martin Uribe and Urban Jermann for many useful suggestions. All errors remain my own. arellano@econ.umn.edu

2 1 Introduction Emerging markets tend to have volatile business cycles and experience economic crises more frequently than developed economies. Recent evidence suggests that this may be related to cyclical changes in the access to international credit. In particular, emerging market economies face volatile and highly counter-cyclical interest rates, usually attributed to counter-cyclical default risk. 1 In addition, interest rates and default risk are also systematically correlated with real exchange rate devaluations: devaluations increase the probability of default and default increases the probability of devaluations. The exposure of the economy to such devaluations is magnified by high levels of liability dollarization - that is, "dollar" denominated debt, leveraged to a large extent on domestic income and assets. 2 Figure (1) illustrates these correlations by plotting aggregate consumption, real exchange rates, interest rates and the current account for Argentina. 3 In December 2001, Argentina defaulted on its international debt and fell into a deep economic crisis. During the crisis, consumption and real exchange rates collapsed, interest rates increased, and the current account experienced sharp reversals. 4 This evidence indicates that a priority for theoretical work in emerging markets macroeconomics must be understanding markets for international credit and in particular the joint analysis of default risk, interest rates and real exchange rates. This paper develops a stochastic general equilibrium model with endogenous default risk. The model allows the study of the relation between interest rates, real exchange rates and output, shedding light on potential mechanisms generating the co-movements described above. The terms of international loans are endogenous to domestic fundamentals and depend on incentives to default. The paper extends the approach developed by Eaton and Gersovitz (1981) in their the seminal study on international lending, by analyzing how endogenous default probabilities and fluctuations in output are related. In addition, the model considers a default penalty that is limited to temporary financial autarky, and introduces a nontradable sector to analyze dynamics of interest rates and real exchange rates. The paper characterizes the equilibrium country interest rate schedule and its relation with output, and provides conditions under which default can be an equilibrium outcome. In a quantitative exercise the model is applied to analyze the default experience of Argentina. The model can account for 1 Neumeyer and Perri (2001) and Uribe and Yuen (2003) document the counter-cyclicality of country interest rates for Argentina, Brazil, Ecuador, Mexico, Peru, Philippines, and South Africa. 2 Reinhart (2002), Calvo and Reinhart (2000), Hausmann et. al (2001) document the high levels of liability dollarization in emerging markets foreign debt and its effect on crises and devaluations. 3 Consumption and real exchange rate data are log, current account data are reported as percentage of output. Consumption and current account data are filtered with a linear trend. 4 Mendoza (2002) documents similar dynamics for all the emerging market crises in the 1990 s. 2

3 Argentina Country Interest Rate (left axis) Consumption Real Exchange Rate Current Account Q1 1999Q1 2000Q1 2001Q1 2002Q Figure 1: the recent default in Argentina and for the sharp aggregate dynamics in the default episode. The model in this paper accounts for the empirical regularities in emerging markets as an equilibrium outcome of the interaction between risk-neutral competitive creditors and a risk averse borrower that has the option to default. The borrower is a benevolent government of a small open economy with two sectors, tradables and nontradables. The government in this economy can buy or sell tradable denominated bonds with foreign creditors. The model starts from the premise that default probabilities are endogenous to the economy s incentives to default and they affect the equilibrium interest rate the economy faces. Default is an equilibrium outcome of the model because the asset structure is incomplete since it includes only one period discount bonds that pay a non-contingent face value. Asset incompleteness is necessary in this framework to study time-varying default premia due to equilibrium default. With non-contingent assets, risk neutral competitive lenders are willing to provide debt contractsthatinsomestateswillresultindefaultbychargingahigherpremiumonthese contracts. In addition to more closely reflecting the actual terms of international contracts where foreign debt is largely dollar denominated and contracted at non-contingent interest rates, this market structure has the potential to deliver counter-cyclical default risk, since repayment of non-contingent, nonnegotiable loans in low output, low consumption times is more costly than repayment in boom times. The existence of nontradable goods is important because even though foreign debt cannot be used to smooth consumption of nontradable 3

4 goods, nontradable output fluctuations affect repayment incentives and equilibrium interest rates through changes in the real exchange rate. In the first part of the paper, a simpler version of the model with i.i.d. shocks is considered in order to characterize analytically the equilibrium properties of credit markets and in particular the conditions under which default is an equilibrium outcome. It is shown that default is more likely the lower the output of tradables. This result contrasts with standard participation constraint models that have a complete set of assets, which have the feature that default incentives are higher in good times. The key intuition for why asset market incompleteness can reverse this outcome is that after a long series of low endowment shocks, an economy with incomplete markets can experience capital outflows in bad times. Risk averse agents with large debt holdings that can experience capital outflows have more incentives to default in times of low shocks. In essence, in times of low output, the asset available to the economy does not help agents smooth consumption, thus it is not very valued and default can be preferable than repayment. A similar intuition holds for the relationship between the level of nontradable output and default, although here it depends on how nontradable shocks affect the marginal utility of tradable consumption. We find that default is more tempting for low nontradable shocks when a low nontradable shock increases the marginal utility of tradable consumption. When this condition holds, capital outflows are more costly for lower nontradable shocks and thus default more likely. In the quantitative part of the paper, the model is calibrated to Argentina to study its recent default episode. The model can account for the recent default of Argentina and can match the observed business cycle correlations. Specifically, it can account for the negative correlations between output and consumption with interest rates. It also matches the positive correlation between real exchange rate devaluations and interest rates. The model matches the data in that before a default occurs, the economy faces high interest rates and features sharpreductionsincapitalinflows, sharp reversals from large current account deficits into much smaller deficits or even surpluses, and collapses in consumption and real exchange rates. The relative sizes of the tradable and nontradable sectors are also very important. We find that relatively large nontradable sectors reduce the economy s commitment for debt repayment and further limit the financial integration of economies. A model with only a tradable sector would face much lower equilibrium interest rates and default would be a much less likely outcome. The paper is related to several studies from different strands of the macroeconomics and finance literatures. Some papers have looked at the relation between interest rates and business cycles. Neumeyer and Perri (2001) model the effect that exogenous interest rate 4

5 fluctuations have on business cycles and find that interest rate shocks can account for 50% of the volatility of output. Uribe and Yue (2003) construct an empirical VAR to uncover the relationship between country interest rates and output, and then estimate with a theoretical model this relationship. They find that country spreads explain 12% of movements in output, and that output explains 12% in the movements of country interest rates. These papers, however, do not model endogenous country spreads responding to default probabilities. The debt contractual arrangement in this paper is related to the optimal contract arrangements in the presence of commitment problems such as the analysis by Kehoe and Levine (1993) and Kocherlakota (1996). These studies, however, assume that a complete set of contingent assets are available and search for allocations that are efficient subject to a lack of enforceability. Alvarez and Jermann (2000) show that in this framework each state-contingent asset is associated with a state-contingent borrowing limit. This limit is such that in each state the borrowing country prefers to repay loans rather than default. While it is useful to be able to characterize allocations under the efficiency benchmark, this market structure may not be useful as a framework for understanding actual emerging markets. First, default never arises in equilibrium, so default risk premia are never observed. Second, default incentives in this class of models are typically higher in periods of high output, which is when efficiency dictates loan repayment. These features put these models at odds with the empirical evidence regarding default risk in emerging markets where bond yields are counter-cyclical and where debt prices reflect largely the risk of default. This paper delivers the correct empirical prediction because it assumes an incomplete set of assets where default occurs with a positive probability. In this regard the paper is specially related to the analysis on unsecured consumer credit with the risk of default by Chaterrjee, Corbae, Nakajima and Rios Rull (2002) where they model equilibrium default in an incomplete markets setting. In some recent work Aguiar and Gopinath (2004) introduce shocks to the growth rate to a model similar to one developed here and find that these shocks can help account for the positive correlation between current account and interest rates at the cost of generating acyclical interest rates. Other authors have developed models to study liability dollarization and have concentrated on how the level and volatility of output and relative price of nontradables affect the ability to pay dollar debt. Calvo (1998) shows how collapses in the price of nontradables due to constraints on tradable denominated debt can lead to crises by limiting the ability for debt repayment. Mendoza (2002) examines a stochastic environment where households face a liquidity requirement borrowing constraint (in which debt cannot be larger than an exogenous fraction of current income) to study emerging markets crises. He shows how crises can be the outcome of the equilibrium dynamics of an economy with imperfect credit mar- 5

6 kets. In these models though, the borrowing constraints do not guarantee that debtors have incentives to repay debts. In addition, they do not model the endogeneity of interest rates and the relation between interest rates and liability dollarization. The focus in this paper is on understanding how the level and volatility of tradable and nontradable output affect incentives to default and thus equilibrium country interest rates in an environment of liability dollarization. Results match the empirical facts in that default incentives are higher when the economy has large debt positions, is in recession, and has relatively small tradable sectors. The paper is organized as follows: section 2 presents the theoretical model, section 3 characterizes the equilibrium, section 4 assesses the quantitative implications of the model in explaining the data, and section 5 concludes. 2 The Model Economy Consider small open economy with two sectors, a tradable sector and a nontradable sector. A set of one period discount bonds is available to the government of the small open economy at contingent prices. Debt contracts are not enforceable as the government can choose to default in its debt contracts if it finds it optimal. If the government defaults in its debts, it is assumed to be temporarily excluded from international financial markets and that a portion of the aggregate endowment is lost during the autarky periods. The economy trades discount bonds with risk neutral, competitive foreign creditors. Households are identical and have preferences given by X E 0 β t u(c T t,c N t ) (1) t=0 where 0 <β<1 is the discount factor, c T and c N are consumption of the tradable and nontradable goods, and u( ) is strictly concave and increasing, and twice continuously differentiable. Households receive stochastic endowment streams of tradable y T and nontradable y N goods. The exogenous state vector of sector-endowment shocks is defined as y (y T,y N ) Y. ShocksareassumedtohaveacompactsupportandtobeaMarkovprocessdrawnfroma probability space (y, B(y)) with a transition function f(y 0 y). In addition households receive a transfer of tradable goods from the government in a lump sum fashion. Households trade in the spot market tradable and nontradable goods with p N being the relative price of nontradable goods. 6

7 The government is benevolent in that its objective is to maximize the utility of households. The government has access to international financial markets, where it can buy one period discount bonds B 0 denominated in terms of tradables at price q. The government also decides whether to repay or default on its debt. The bond price function q(b 0,y) depends on the size of the bond B 0 and on the aggregate shock y because default probabilities depend on both. A purchase of a discount bond with a positive value for B 0 means that the government has entered into a contract where it saves q(b 0,y) B 0 units of period t tradable goods to receive B 0 0 units of tradable goods the next period. A purchase of a discount bond with negative face value for B 0 means that households have entered into a contract where they receive q(b 0,y) B 0 units of period t tradable goods, and promise to deliver, conditional on not declaring default, B 0 units of tradable goods the next period. The government rebates back to the households all the proceedings from its international credit operations in a lump sum fashion. Whenthegovernmentchoosestorepayitsdebts,thetradablesectormarketclearing condition is the following: c T = y T + B q(b 0,y)B 0 The market clearing condition for the nontradable sector, requires that nontradable consumption equals nontradable output at all times. c N = y N Given that the government is benevolent, it effectively uses international borrowing to smooth consumption. But the idiosyncratic income uncertainty induced by y T and y N cannot be insured away with the set of bonds available to the economy, which pay a time and state invariant amount of tradable goods only. That is, asset markets in this model are incomplete not only because of the endogenous default risk, but also because of the set of assets available. If the government defaults, it is assumed that current debts are erased and that it is not able to save or borrow. The government will remain in financial autarky for a stochastic number of periods. There is an exogenous constant probability θ that the government can reenter the market. This is a simple way to model that governments that default on their international debt lose access to international financial markets only temporarily. In addition, in the period when the government defaults, endowments fall a proportion (1 λ), where λ 1. The assumption that default may reduce output can be rationalized by the common 7

8 view that after default there is a disruption in the countries ability to engage in international trade, and this reduces the value of output (Cole and Kehoe 2000, Conklin 1998, Obstfeld and Rogoff 1989). When the government chooses to default market clearing conditions require that consumption equals output : c T = λy T c N = λy N Foreign creditors have access to an international credit market in which they can borrow or lend as much as needed at a constant international interest rate r>0. Creditors have perfect information regarding the economy s endowment processes and can observe each period the endowment levels. 5 Creditors are assumed to be risk neutral and behave competitively. Creditors engage in Bertrand competition, offering contracts to the government that gives them expected zero profits. They choose loans B 0 to maximize expected profits φ, taking as given prices: φ = qb 0 (1 δ) + 1+r B0 (2) where δ is the probability of default. For positive levels of foreign asset holdings, B 0 0, the probability of default is zero and thus the price of a discounted bond will be equal to the opportunity cost for investors. For negative asset holdings, B 0 < 0, the equilibrium price accounts for the risk of default that creditors are facing, that is, the price of a discount bond will be equal to risk adjusted opportunity cost. 6 This requires that bond prices satisfy: q = (1 δ) 1+r (3) The probability of default δ is endogenous to the model and depends on the government incentives to repay debt. Since 0 δ 1, the zero profit requirement implies that bond prices q lie in the closed interval [0, (1 + r) 1 ]. The country gross interest rate can be interpreted 5 We assume for simplicity that each period the government enters into a debt contract with only one creditor. This can be generalized by assuming that creditors can observe previously written contracts and contracts that are written firsthavetobehonoredfirst. 6 Risk adjustment in this framework is not due to compensation for risk aversion as lenders are risk neutral. It reflects the risk neutral compensation for a lower expected payoff. 8

9 as the inverse of the discount bond price, 1+r c = 1 q. The timing of decisions within each period is as follows The government starts with initial assets B, observes the endowment shock y, and decides whether to repay its debt obligations or default. If the government decides to repay, then taking as given the schedule bond price q(b 0,y) the government choose B 0 subject to the resource constraints. Then creditors taking as given q choose B 0. Finally consumption of tradable and nontradable goods c T,c N takes place. 3 Recursive Equilibrium We define a recursive equilibrium in which the government does not have commitment and in which the government, foreign creditors and households act sequentially. Given aggregate states s =(B,y), the policy functions for the government B 0, the price function for bonds q and the policy functions for the consumers c T,c N together with the relative price p N determine the equilibrium. Households problem is static. They observe their endowment shocks and taking as given the aggregate states, government transfers and the nontradable price they choose optimal consumption. Their first order condition equates the marginal rate of substitution between nontradable and tradable consumption to the relative price. u c N (c T,c N ) u c T (c T,c N ) = pn (4) Given that purchasing power parity (PPP) is assumed to hold for the tradable sector, the real exchange rate for this small economy is the domestic consumption price index p c which is a function of the nontradable price, p N, and the tradable price, which is normalized to 1. Foreign creditors in the model are risk neutral and behave competitively. They lend the amount of bonds wanted by the government as long as the price satisfies q(b 0,y)= (1 δ(b0,y)) 1+r Foreign creditors in the model are pretty passive, they lend bonds as long as the gross return on the bonds equals 1+r. The government observes the aggregate endowment shocks y, and given initial foreign assets B, choose whether to repay or to default. If the government chooses to repay its debt obligations and remain in the contract, then it chooses the new level of foreign assets B 0. 9

10 The government understands that the price of new borrowing q(b 0,y) depends on the states y and on its choice of B 0. The government also understands that its choices will affect the households choices c T,c N and p N and internalizes the domestic market clearing conditions. Thegovernmentobjectiveistomaximizethelifetimeutilityofhouseholds. Define v o (B,y) as the value function for the government who has the option to default, and that start the current period with assets B and endowment shocks y. The government decides on whether to default or repay its debts, to maximize the welfare of households. Note that only when the government has debt (i.e. negative assets) could the default option be optimal. Given the option to default, v o (B,y) satisfies: v o (B,y) =max v c (B,y),v d (y) ª (5) {c,d} where v c (B,y) is the value associated with not defaulting and staying in the contract and v d (y) is the value associated with default. When the government defaults, the economy will be in temporary financial autarky; θ is the probability that it will regain access to international credit markets. The value of default is given by the following: Z v d (y) =u(λy T,λy N )+β θv o (0,y 0 )+(1 θ)v d (y 0 ) f(y 0 y)dy 0 (6) y 0 If the government defaults endowments fall, and consumption equal output. When the government chooses to remain in the credit relation, the value conditional on not defaulting is the following ½ Z ¾ v c (B,y) =max u(y T q(b 0,y)B 0 + B,y N )+β v o (B 0,y 0 )f(y 0 y)dy 0 (7) (B 0 ) y 0 The government decides on optimal policies B 0 to maximize utility. The decision to remain in the credit contract and not default is a period by period decision. That is, the expected value from next period forward incorporates the fact that the government could choose to default in the future. The government also faces a lower bound on debt, B 0 Z, which prevents the government from running ponzi schemes but is otherwise not binding in equilibrium. The government default policy can be characterized by default sets and repayment sets. Let A(B) be the set of y 0 s for which repayment is optimal when assets are B such that 7 : 7 We assume that if the government is indifferent between repayment and default, default is chosen. 10

11 A(B) = y Y : v c (B,y) >v d (y) ª and D(B) = ea(b) be the set of y 0 s for which default is optimal for a level of assets B, such that: D(B) = y Y : v c (B,y) v d (y) ª (8) The value function can then be represented more precisely by the following dynamic problem where the government decides on optimal borrowing taking into account that its choice on assets may imply defaulting in some states. v c (B,y) = ½ Z Z 0 ¾ max u(y T q(b 0,y)B 0 + B,y N )+β v c (B 0,y 0 )f(y 0 y)dy 0 + v d (y 0 )f(y 0 y)dy (B 0 ) A(B 0 ) D(B 0 ) The above centralized government borrowing problem can be decentralized in multiple ways, with the simplest being lump sum taxation as presented here. In a separate appendix, it is shown that the above problem can also be decentralized as in Kehoe and Perri (2004) by letting households borrow directly from foreign creditors. The government in this case makes the economy wide default decision, and levies a savings tax that gives households the right incentives for holding the optimal level of debt. For simplicity in the exposition we have assumed here that the government has access to lump sum taxes. Having developed the problem for each of the agents in the economy, the equilibrium is defined. Let s = {B,y} be the the aggregate states for the economy. Definition 1. The recursive equilibrium for this economy is definedasasetofpolicy functions for (i) consumption c T (s) and c N (s), nontradable price p N (s) (ii) government s asset holdings B 0 (s), repayment sets A(B) and default sets D(B), and (iii) the price for bonds q(s, B 0 ), such that: 1. Taking as given the government policies, policy functions for households c T (s), c N (s), and the relative price p N (s) satisfy the households optimization problem and domestic market clearing conditions hold. 2. Taking as given the bond price function q(b 0,y), the government s policy functions B 0 (s), repayment sets A(B) and default sets D(B), satisfy the government optimization problem. 11

12 3. Bonds prices q(b 0,y) are such that they reflect the government default probabilities and they are consistent with creditor s expected zero profits such that the loan market clears. The equilibrium bond price function q(b 0,y) has to be consistent with government s optimization and with expected zero profits for foreign creditors. That is, q correctly assesses the probability of default of the government. 8 Default probabilities δ(b 0,y) and default sets D(B 0 ) are then related in the following way: Z δ(b 0,y)= f(y 0 y)dy 0 (9) D(B 0 ) When default sets are empty, D(B 0 )=, equilibrium default probabilities δ(b 0,y) are equal to 0. That is, the economy with assets B 0 never chooses to default for all realization of the endowment shocks. When D(B 0 )=Y, default probabilities δ(b 0,y) are equal to 1. More generally, default sets are shrinking in assets, as the following proposition shows: Proposition 1. (Default sets are shrinking in assets) For all B 1 B 2,ifdefaultis optimal for B 2, in some states y, then default will be optimal for B 1 for the same states y, that is D(B 2 ) D(B 1 ). Proof. See Appendix. This result is similar to Eaton and Gersovitz (1981) and Chatterjee, et al. (2002). Intuitively, the result follows from the property that the value of staying in the contract is increasing in B, and that the value of default is independent of B. As assets decrease, the value of the contract monotonically decreases, while the value of default is constant. Thus, if default is preferred for some level of assets B, for a given state y, the value of the contract is less than the value of default. As assets decrease, the value of the contract will be even lower than before, and so default will continued to be preferred. Since stochastic shocks are assumed to have a bounded support and the value of the contract is monotonically decreasing as assets fall, there exists a level of B that is low enough, such that for all endowment shocks default is optimal and default sets are equal to the entire endowment set. On the other hand, given that default can only be preferable when assets have a negative value (i.e. when the government is holding debts), there exists a level of assets B 0, such that default sets are empty. These two properties of default sets can be summarized as follows. 8 Chatterjee et al (2001) prove the existence of an equilibrium price schedule in a similar environment for their work on consumer default risk. We conjecture that the existence proof for this model follows that of Theorem 4 in their paper. 12

13 Definition 2. Denote B the upper bound of assets for which the default set constitutes the entire set and let B be the lower bound of assets for which default sets are empty, where B B 0 due to Proposition 1. B =sup{b : D(B) =Y } B =inf{b : D(B) = } For asset holdings greater than B, default is never optimal for all y and equilibrium bond prices are equal to (1 + r) 1 because default probabilities are zero. 9 For asset holdings B B default is always optimal and equilibrium prices for these bonds are zero because default probabilities are one. Given that default sets are shrinking in the level of assets, condition (9) implies that equilibrium default probabilities are decreasing in B 0, and the equilibrium price function q(b 0,y) is an increasing function of B 0. Lower levels of assets will be associated with larger default probabilities, and thus discount prices for those bonds will be lower to compensate risk neutral investors for a lower expected payoff. That is, larger loans are generally more expensive. Equilibrium bond prices are also contingent on the endowment shocks, because the probability distribution from which shocks are drawn the next period depends on today s shocks. Since the risk of default varies with the level of debt and depends on the stochastic structure of the endowment shocks, competitive risk neutral pricing requires that equilibrium bond prices depend on both B 0 and y. 3.1 Case of i.i.d. Tradable Endowment Shocks This section characterizes the equilibrium bond price function and the default decision for the case of a constant nontradable endowment and i.i.d. tradable endowment shocks. When endowment shocks are i.i.d., equilibrium bond prices are independent of the endowment realization and are only a function of the level of assets q(b 0 ) because today s shock gives no information on the likelihood of tomorrow s shock. We will assume that λ =1,nooutput loss in autarky, and θ =0, financial autarky is permanent after default. The results can be generalized for other parameters of λ and θ. Proposition 2. Default incentives are stronger the lower the tradable endowment. For all y1 T <y2 T,if y2 T D(B), theny1 T D(B). 9 Zhang (1997) introduced this debt limit as the no default debt limit in his work on participation contraints under incomplete markets. 13

14 Proof. See Appendix. This result comes from the property that utility is increasing and concave in tradable consumption and by noting that default can be optimal only if under no default the economy experiences net capital outflows (B q(b 0 )B 0 < 0). In fact, when for some state default is optimal, there are no contracts available to the government such that the economy can experience capital inflows given that level of debt for all states. Given that utility is increasing and concave in consumption, and that the economy is not able to borrow more when it has the low endowment, repayment is more costly in this low endowment state and thus default more likely. Intuitively, the asset available to the economy is not a very useful insurance instrument for a highly indebted economy, because in times of a low endowment it cannot raise enough resources to smooth consumption. Thus, the asset the economy is giving up is not very valuable and default may be preferable in times of low endowments. Endowment shocks have generally two opposing effects on default incentives. When output is high, the value of default is relatively high, increasing default incentives. But at the same time, when output is high, the value of repayment is relatively higher too, decreasing default incentives. With an incomplete set of assets, and i.i.d. shocks, the latter effect dominates and thus default is more likely the lower the tradable endowment. This result contrasts with the participation constraint models that have a complete set of contingent assets. These models have the feature that default incentives are higher in times of good endowment shocks. In fact, for small open economy models with participation constraints and a complete set of contingent assets, default incentives are always higher in the good states because only autarky is affected by the current endowment, as the value of staying in the contract is constant in the long run and independent of the economy s specific endowment. Proposition 3. If default sets are non-empty, then they are closed intervals where only the upper bound depends on the level of assets: D(B) = [y,y (B)] for B B (10) where y (B) is a continuous, non-increasing function of B, such that: ( y y (B) :v d (y (B)) = v c (B,y (B)) for B B B (B) = y for B<B ) (11) Proof. See Appendix. 14

15 Proposition 3 proves that for B [B, B], there will be a unique y where the contract value and continuation value cross. Default sets can then be characterized solely by a closed interval where only the upper bound is a function of the level of assets. For a given a level of assets B [B, B], default will be optimal for endowment levels less than or equal to y (B), and repayment will be optimal for endowment levels greater than y (B). The function y (B) is the default boundary that divides the y, B space into the repayment and default regions. 10 The first order condition of the government with respect to asset holdings can be presented more sharply in this case by the following condition: [q(b 0 )B 0 Z ] u B 0 c T = β vb c 0(B0,y 0 )f (y 0 ) dy 0 (12) A(B 0 ) Equation (12) equates the marginal utility of consumption today to the expected marginal value of wealth tomorrow for the states where repayment is optimal. The marginal cost from borrowing a loan of size B 0 in the current period is the expected marginal disutility of consumption from repaying that loan the following period. As opposed to standard intertemporal conditions for models without default, here that cost is only experienced if in the following period the government choose to repay. That is, the cost of repaying is realized for the set of y s, A(B 0 )=(y (B 0 ), y], for which repayment is optimal when the economy has assets equal to B 0. Given that default sets are such that only the upper bound depends on the level of debt, the equilibrium price function q(b) can then be characterized by the following condition: q(b) = 1 1+r [1 F (y (B))] (13) where F is the cumulative probability distribution of the stochastic endowments. Equilibrium bond prices will fall in three ranges. For asset levels above B prices are equal to the inverse of the risk free rate. For asset levels less than B, prices are zero. For intermediate asset levels, B [B, B] prices will be increasing in the level of assets because y (B) is decreasing in this range. Note that the bond price function will be crucially dependent upon the probability distribution of the endowment shocks. If initial bond values are B then the probability of default at every point of the state space is given by F (y (B)), whichisgreaterthanzeroinforb [, B]. The next issue to be addressed is whether the economy would ever choose a B 0 such that D(B 0 ) 6=. That is if in 10 In countinuous time optimal stopping problems, it has been shown that under special cases not only the function is continuous at the boundary, but present a smooth pasting condition. Which would imply that the derivatives with respect to y of the continuation and default values are equal at the boundary. 15

16 q(b)b Risky Borrowing slope=(1+r) -1 B B* B B Figure 2: Total Resources Borrowed the ergodic distribution of assets a point exists where default has a positive probability, such that if initial bond holdings are larger than B, the model can have default as an equilibrium outcome. To clarify this issue, imagine the economy happens to start at B B, then in that period the economy would default with probability one. But given that discount prices are zero in this range, if the economy s initial bond holdings are greater than B, the economy would never choose as optimal asset holdings levels of B 0 B, becauseitwouldgetnothing this period, and would incur a liability the following period. The range of B 0 for which default can potentially be an equilibrium outcome is limited to (B, B], becausehereiswhere default sets are non-empty and equilibrium prices are different from zero. Intuitively, the necessary condition for default to be an equilibrium outcome of the model is that the equilibrium price function does not decrease "too fast" as assets decrease. For default to potentially be an equilibrium outcome, the economy must be able to find it optimal to borrow bonds less than or equal to B such that the economy is exposed to the risk of default. And for this to ever be an optimal decision, the economy should be able to increase total resources borrowed q(b)b, that is, have a higher level of consumption, by choosing a lower level of assets at a lower price. This means that the equilibrium total resources borrowed q(b)b needs to be increasing for some B (B, B). Figure (2) helps visualize this issue. 16

17 The slope of q(b)b for B>B equals (1 + r) 1 because bond prices are constant and default probabilities are zero. The slope of q(b)b for B < Bis equal to zero, because bond prices are zero. For the intermediate range (B, B) the slope at equilibrium prices is: [q(b)b] B = 1 (1 + r) ¾ ½[1 F (y (B))] f((y (B)) B y B Note that the sign of this derivative is ambiguous because bond positions B (B, B) are negative, and y (B) is decreasing in B. 11 In fact if there exists some B (B, B) for which [q(b )B ] =0this level of assets corresponds to the endogenous borrowing limit in B the model. The government would never find optimal to choose a level of assets below B becauseitcanalwaysfind some other contract such that consumption the current period increases by the same amount while incurring a smaller liability for next period. The region that is relevant for risky borrowing and thus for default to be an equilibrium outcome of the model is then B (B, B). The necessary condition for the above derivative to be positive within a range depends on f(y) the hazard function (i.e. ) of the probability distribution in the neighborhood of [1 F (y)] y relative to how fast the upper bound on the default sets increase with debt. The following proposition summarizes these findings: Proposition 4. Default can be an equilibrium outcome of the model for all probability distributions over y, y satisfying the property: 1 lim B B h(y (B)) > lim y (B) B B B B where h(y) is the hazard function of the distribution. Proof. When the above condition holds, [q(b)b] > 0 in the neighborhood of B from B the left. Given that y (B) is continuous by Proposition 3, total resources borrowed increase for lower levels of assets in the region where assets carry a default premium. The hazard function h(y (B)) represents the instantaneous probability of default at B for B B. The above condition requires that the instantaneous probability of default as B approaches B from the left is sufficiently small such that as B decreases (debt increases), the total resources borrowed increase. Due to Proposition 3, y (B) =y, and thus the condition is a restriction on the probability of the bad endowment shock. The government might then (14) 11 In general q(b)b may not differentiable at the points B and B. 17

18 be willing to take on a risky loan, because in periods of low endowments it can increase consumption. Asufficient condition for default to be an equilibrium outcome of the model is the following: Corollary 4.1. Default can be an equilibrium outcome of the model for all probability distributions over y, y satisfying the property: lim y y h(y) =0 where h(y) is the hazard function of the distribution. Proof. See Appendix. For all probability distributions satisfying the above property, the model will present a region in the state space where engaging in risky borrowing can increase consumption or [q(b)b] > 0, making default a positive probability event. B 3.2 Case of i.i.d Nontradable Endowments Now the role of volatile nontradable endowment is explored and its effect on default incentives. Here it is assumed that nontradable endowments follow an i.i.d. stochastic process, and without loss of generality that λ =1, and θ =0. Proposition 5. Default incentives are stronger for low nontradable endowments if the cross derivative of the utility function is negative. For all y1 N y2 N,if y2 N D(B), then y1 N D(B) if u c T c < 0. N Proof. See Appendix. Given that assets are tradable denominated only, nontradable fluctuations affect default decisions by their effect on the utility of tradable consumption. When u c T cn < 0, alow nontradable shock will tend to increase the marginal utility of tradable consumption, giving the government incentives for borrowing. In the region of default the economy experiences capital outflows and the fact that households cannot borrow as much as desired because of high interest rates and debt limits is relatively more costly for households who experience a low nontradable endowment if the above condition holds. Thus, default is more likely in the low nontradable states because repayment of tradable denominated loans is more costly. 18

19 For CES preferences it is well known that sign of this cross derivative depends on the relative magnitudes of the elasticity of intratemporal substitution between tradables and nontradables versus the intertemporal elasticity of substitution (see for example Obstfeld and Rogoff 1996 textbook). When the elasticity of intratemporal substitution between tradables and nontradables is greater the intertemporal elasticity of substitution the cross derivative is negative, and otherwise its positive. Intuitively, decisions on optimal asset holdings depend on nontradable output fluctuations due to the effect on the relative price of nontradables. The intertemporal elasticity through time and intratemporal elasticity between tradable and nontradable consumption pull the time path of nontradable prices in opposite directions. When the intratemporal elasticity is greater than the intertemporal elasticity a low nontradable shock today tends to produce a decreasing time path in the nontradable price, which givesincentivesforborrowinginthecurrentperiod. The analytical characterization of the equilibrium in this section is for i.i.d. shocks, with bond prices not depending on the endowments realization and only increasing in the level of assets demanded. However given that debt is used for insurance purposes, the policy function for assets is increasing in the endowments as in Hugget (1993), so that when the economy is hit by negative endowment shocks, more debt is demanded. This generates in the time series a negative correlation between interest rates and endowments even for i.i.d. shocks because higher debt is associated with higher interest rates. The following section analyzes the relation between interest rates and output for a persistent endowment process and where the negative relation between output and interest rates remains. 4 Simulations 4.1 Data for Argentina In December of 2001 Argentina experienced one of the largest defaults in recent history, defaulting on $100 billion dollars of their external government debt which represented 37% of Argentina 2001 GDP. It also experienced a severe economic crises with output decreasing about 20% percent at the time of the default. This section documents some statistics of the Argentinian economy corresponding to the period of default. Table 2 presents business cycle statistics for Argentinian data that are quarterly real series seasonally adjusted taken from the Ministry of Finance (MECON) from 1993 to 2001 and filtered with a linear trend 12. The interest rate series are the EMBI index for Argentina 12 The series are constructed such that all variables are consistent with the model s statistics. Specifically, 19

20 which is taken from the dataset from Neumeyer and Perri (2004) and MECON. Output and consumption data are log, and the current account data are reported as a percentage of output. Real exchange rates are constructed from dollar nominal exchange rates (dollar per peso) using the ratio of the consumer price index for Argentina and the US. The table also presents the percent deviation of the variables in Q1 2002, the default period. Table 1. Statistics for Argentina Default episode Prior default Q Q x : Q std(x) corr(x, y) corr(x, r) Interest Rates Spread Consumption Tradable Output Nontradable Output Aggregate Output Current Account Real Exchange Rate Aggregate output, tradable output and nontradable output are all negatively correlated with interest rates. This negative relations are much stronger in the default episode because in the crisis output plummeted and interest rates skyrocketed. Consumption is more volatile than nontradable output, less volatile than tradable output, and negatively correlated with interest rates. This negative relation is also magnified in the default episode. Real exchange rates prior to the default episode were extremely stable in this period, and slightly negatively correlated with interest rates. But in the default episode real exchange rates collapsed from 1 dollar per peso, to 0.4 dollars per peso or over 70% while interest rates spike by 20% giving a negative relation between devaluations an interest rates. The current account is countercyclical which is accentuated in the time of the default because Argentina experienced a sharp current account reversal during the crisis as foreign credit dried up. The countercylicality of the current account is due to the negative relation between the current account and nontradable output. The correlation between nontradable output and the current account is negative and equal to -0.73, while the correlation between the current account and tradable output is positive and equal to Interest rates in Argentina are slightly less volatile than nontradable output, and negatively correlated with output and consumption. The mean spread in Argentina defined as aggregate output is denominated in terms of tradables and sectoral output and aggregate consumption is denominated in real terms deflated by the appropriate price deflator. 20

21 the difference in the US annual 5 year maturity bond yield and the Argentina EMBI is 7.38% in this period. As the table shows all variables experienced very dramatic deviations at the time of the default. 4.2 Calibration and functional forms The model is solved numerically to evaluate its predictions regarding accounting for default episodes and aggregate dynamics in crisis, and its quantitative implications for the business cycle properties of interest rates, real exchange rates and consumption. The main issue of interest in the quantitative analysis is to address whether adding an endogenous default decision to a very simple endowment model can help account for the real dynamics observed in emerging markets in times of default and crises. The benchmark model is calibrated to match certain features of the Argentinian economy. 13 The following utility function is used in the numerical simulations: u(c T t,c N t )= c(ct t,c N t ) 1 σ, 1 σ where c(c T t,c N t ) is the constant elasticity of substitution aggregator h ω c T η i t +(1 ω) c N η 1 η t. The parameters of the benchmark model are calibrated to mimic some of the empirical regularities in the Argentinian economy or taken from other emerging markets studies. For the preference parameters, the risk aversion coefficient is set to 5 which is a common value use in real business cycles studies for emerging markets. The elasticity of substitution between tradable and nontradable consumption 1/(1+µ) is taken from Gonzales and Neumeyer (2003), where they estimate the elasticity for Argentina to be equal to To calibrate the relative sizes of the tradable and nontradable sectors in Argentina, we follow the standard methodology of assessing the degree of tradability of goods by computing the share of total trade (exports plus imports) of each sector as a percentage of total sectoral gross output. We find that the agricultural, manufacturing and energy sectors have a high degree of tradability, with an average share in this period of 0.38, 0.78 and 0.34 respectively. 13 The model is solved by a value function iteration algorithm that allows for the bond price vector to be endogenous. Specifically, endogenous and exogenous states are discretized, and the model is solved by iterating on the value function for an intial guess of the bond price vector. The bond price vector is updated with a Gauss-Seidel algorithm, using the creditors equilibrium zero profit condition. The procedure is repeated until the convergence criterion is met for the bond price vector. 21